The options I've seen so far are to configure SharePoint to use Shibboleth, CAS, InCommon, or something like that; or to host the ASP.NET applications on the same server as SharePoint 2010 and have them use the same Forms authentication ticket.

Would it be possible to host an ASP.NET application within SharePoint so that it can directly check to see if someone is logged in to SharePoint or do something similar? Are there any other methods that can be used?